J.Appl. Phys. 83, 5410

JOURNAL OF APPLIED PHYSICS

VOLUME 83, NUMBER 10

15 MAY 1998

Excitons in near-surface quantum wells in magnetic fields: Experiment and theory N. A. Gippius,a),b) A. L. Yablonskii, A. B. Dzyubenko, and S. G. Tikhodeev General Physics Institute, Russian Academy of Sciences, 117942 Moscow, Russia

L. V. Kulik and V. D. Kulakovskii Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russia

A. Forchel Technische Physik, Universita¨t Wu¨rzburg, D-97074 Wu¨rzburg, Germany

~Received 5 January 1998; accepted for publication 6 February 1998! The exciton transition and binding energies have been investigated in near-surface InGaAs/GaAs quantum wells theoretically and experimentally ~by photoluminescence and photoluminescence excitation spectroscopy at 4.2 K!. The contribution induced by vacuum has been analyzed for the ground and excited exciton states in perpendicular magnetic fields up to 14 T. The vacuum potential barrier has been shown to increase the magnetoexciton transition energies, \ v n , but nearly not to influence their binding energies, E n . In contrast, the image charges ~caused by the abrupt, one order of magnitude, decrease of the dielectric constant at the semiconductor-vacuum interface! modify the Coulomb interaction and lead to the increase of both \ v n and E n . The magnetic field has been found to enhance the contribution of the image charges to the exciton binding energy and to decrease their influence on the transition energy. The effect is due to the in-plane exciton wave function squeezing in a magnetic field. © 1998 American Institute of Physics. @S0021-8979~98!01910-0#

I. INTRODUCTION

structures allows one to fabricate such structures with well defined both InGaAs QW and GaAs cap layer thicknesses. Second, both the ground and excited exciton state energies can be measured in available high quality structures with an accuracy providing a quantitative comparison with theoretical calculations. Third, the excitons in these structures are large in comparison to a lattice constant and can be quantitatively described in the Wannier–Mott approximation. In addition, for more detailed studies of the exciton dielectric confinement in such structures,one can apply an external magnetic field. The magnetic field modifies the effective e-h interaction via the change of the exciton wave functions and thereby opens new possibilities to study the dielectric confinement effect in detail. In the present article we have carried out a systematic experimental and theoretical study of the effect of image charges on the exciton properties in the InGaAS/GaAs NSQWs with cap layer thickness, L cap53–20 nm. The theory of NSQWs excitons has been developed allowing to calculate the ground and excited exciton energies with and without taking into account image charges at the semiconductor-vacuum ~SV! interface both for zero and nonzero magnetic fields. This theory includes the previously considered limits of zero10 and strong11 magnetic fields, and it is able to describe the intermediate magnetic field regime. In this work we have considered s- as well as p-excitonic states, and studied the effect of a near-surface perpendicular electric field. The exciton transition energies for the ground and several excited states have been measured in a wide range of magnetic fields B50 – 14 T with the use of the photoluminescence excitation ~PLE! technique. Finally, we have

1,2

It has been predicted that excitons in narrow semiconductor layers surrounded by a dielectric of a smaller dielectric constant have to be significantly enhanced. This effect originates from the modification of the electron-hole (e-h) interaction by the images induced by the interfaces between materials with different dielectric constants. This effect is referred to as the dielectric enhancement of excitons. It has been extensively discussed theoretically,3–7 whereas the experimental study of this effect has met a problem of a fabrication of suitable very narrow ~of a few nm! semiconductor layers in the dielectric environment. The experimental study of the exciton dielectric enhancement has been undertaken in thin CdTe films on a dielectric substrate,8 in self-organized PbI based semiconductor/insulator superlattices, quantum wells ~QWs!, wires, and dots,9 and in near surface InGaAs/ GaAs quantum wells ~NSQWs!, i.e., InGaAs QWs separated from vacuum by a thin ~3–20 nm! GaAs cap ~barrier! layer.10,11 However, very complicated absorption spectra, measured in Ref. 8, and small ~only of the order of few interatomic distances! exciton sizes in PbI based structures,9 did not allow one to carry out systematic studies of the exciton dielectric enhancement in such semiconductor-insulator structures. Up to date the most promising structures for detailed experimental investigation of the exciton dielectric enhancement seem to be the NSQWs. First, a well developed technique of growing and etching of InGaAs/GaAs QW a!

Present address: Technische Physik, Universita¨t Wu¨rzburg, D-97074 Wu¨rzburg, Germany. b! Electronic mail: [email protected] 0021-8979/98/83(10)/5410/8/$15.00

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© 1998 American Institute of Physics

Downloaded 10 Jan 2003 to 128.205.17.59. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

Gippius et al.

J. Appl. Phys., Vol. 83, No. 10, 15 May 1998

analyzed the observed evolution of the s-exciton transition energies with increasing magnetic field and decreasing cap layer thickness. Such an analysis has allowed us to separate the effect on s-excitonic states of the high vacuum potential barrier from that of the image charges and to carry out the detailed study of the dielectric enhancement effect. The article is organized as follows. In Sec. II the theoretical model for magnetoexcitons in NSQWs is presented. In Sec. III the experimental methods are described. In Sec. IV we present the theoretical and experimental data and their comparison.

II. THEORY A. Model equations

We consider strained InGaAs NSQWs with GaAs barriers. The light–heavy hole splitting in such QWs is strongly enhanced due to the strain ~see, e.g., Ref. 12!, therefore we can neglect the split-off light-hole band and consider only heavy-hole magnetoexcitons in a rather wide range of energies. In this approximation, the NSQW electron-hole Hamiltonian in the magnetic field B5(0,0,B) takes the form H5H ez1H hz1H 2D1U eh[H 0 1U eh .

~1!

U eh~ r ,z e ,z h ! 52 1

j5e,h.

~2!

The band-offset potentials are V j 5` ( j5e,h) in vacuum (z,0), V e and V h inside the InGaAs QW (L cap,z,L cap 1L QW), and V j 50 in the barrier GaAs layers, m ez and m hz are the electron and heavy hole effective masses, respectively. The potential V self~ z ! 5

S D

1 e 2 e 21 , 2 e e 11 2 u z u

~3!

~where e is the dielectric constant of the semiconductor! takes into account the repulsion of the charge from its selfimage; we neglect a small difference between e in Inx Ga12x As and GaAs. This approach can be modified to account for an electric field along the z axis, E5(0,0,E), by additional terms in Eq. ~2!, 1(2)eEz e (h). Hamiltonian H 2D describes the two-dimensional ~2D! motion of a free e-h pair in the magnetic field H 2D5

S

1 e 2i\¹ r e 1 Ae 2m e i c 1

S

D

C K50,s ~ re ,rh ! 5exp

S

~4!

where A j 5 12 B3r j is the vector potential in the symmetric gauge, r5 re 2 rh 5(x,y), and r j 5( r j ,z j ). We neglect also a small mass difference in the well and barrier layers. The potential

1 ~ z e 1z h ! 2

i @ r3R# z

f n n~ r ! 5

1 ~ 2 p l 2B ! 1/2

2l 2B

Ln

G

~5!

D

z 1~ z e ! j 1~ z h ! ~6!

S D S D r2 r2 2 exp 2 2l B 4l 2B

and L n (x) is the Laguerre polynomial, n50,1,... . As the experiments show ~see Sec. III!, additional transitions are resolved, which can be ascribed to nominally optically inactive 2 p 6 heavy-hole excitons ~see Sec. IV below!. The wave functions of, e.g., K50 p 6 excitons ~with l z 561! can be written in forms similar to Eq. ~6! ~see also Ref. 17! C K50,p 1 ~ re ,rh ! 5exp

C K50,p 2 ~ re ,rh ! 5exp 3

S

i @ r3R# z 2l 2B

(n A ~np

2

,

Ar

(n A ~ns !f nn~ r ! ,

3

D

1 2

where K is the conserved magnetic center-of-mass momentum,15,16 z 1 and j 1 are the ground state eigenfunctions of Hamiltonians H ez and H hz , respectively, l B 5(\c/eB) 1/2 is the magnetic length, R5(m e i re 1m h i rh )/M , M 5m e i 1m h i

2

1 e 2i\¹ r h 2 Ah 2m h i c

Ar 2 1 ~ z e 2z h ! 2

e 21 e 11

3 \2 ]2 1V j ~ z j ! 1V self~ z j ! , 2m jz ] z 2j

1

takes into account, in addition to the Coulomb interaction between the electron and hole, the attraction of the electron to the hole image, and of the hole to the electron image. In order to solve the quantum-mechanical problem of the magnetoexciton in a NSQW, one can diagonalize numerically the matrix of the Coulomb potential ~5! on the basis of noninteracting two-particle states of Hamiltonian H 0 . Such an approach, when the expansion is performed in Landau levels ~LLs!, has been used previously in the problem of quasi-two-dimensional excitons in coupled double QWs13 in a magnetic field ~see also Ref. 14!. Assuming a strong vertical quantization in the QW potentials V j (z j ), j5e,h, the wave functions of optically active K50 s excitons ~with the angular momentum projection of the relative e-h motion l z 50! take the form

Here H jz52

F

e2 e

5411

S

1!

(n A ~np

2!

z 1~ z e ! j 1~ z h !

f n11 n ~ r! ,

i @ r3R# z 2l 2B

D D

~7!

z 1~ z e ! j 1~ z h !

f n n11 ~ r! ,

~8!

where


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f n11 n ~ r! 5 f n n11 ~ r! * 5

1 ~ 2 p l 2B 2 @ n11 # ! 1/2

S D

3exp 2

F G S D x1iy 1 r 2 Ln lB 2l 2B

r2 , 4l 2B

f n n 8 ( r) are the factored wave functions in a magnetic field, the quantum numbers n and n 8 correspond to the electron and hole Landau level numbers, respectively; the angular momentum projection l z 5n2n 8 . The expansion in Landau levels converges rapidly in high magnetic fields l B /a Be(h) ,1 @ a Be(h) 5 e \ 2 /m e(h) i e 2 # . In the limit l B /a Be(h) !1, the l z 50 excitonic states labeled ~using the 2D hydrogenic notations! ns, n51,2,..., are predominantly formed by the orbital f n21n21 ( r) in the n21 electron and hole Landau levels. Similarly, the l z 561 excitonic states labeled np 1 @and np 2 # are formed predominantly by the orbitals f n n21 ( r) @and f n21 n ( r), respectively#, i.e., in the n @ n21 # electron and n21 @ n # hole Landau levels. At low magnetic fields, a number of other Landau levels are admixed due to the Coulomb e-h interaction. Taking into account of up to 36 Landau level orbitals f n n 8 ( r), we have numerically treated in Ref. 11 the problem of magnetoexcitons in InGaAs/GaAs NSQWs with reasonable accuracy at B.3 T. Such an approach cannot be used at lower magnetic fields, because the number of Landau levels to be taken into account to achieve the needed accuracy increases rapidly.14 In the regime of strong QW confinement, it is possible to overcome this difficulty. Indeed, the Schro¨dinger equation for the K50 exciton wave function, after averaging of Hamiltonian Eq. ~1! over the quantum confined electron and hole eigenfunctions in the z direction, is reduced to an effectively one-dimensional ~ID! equation H r c n,l z ~ r! 5E n,l z c n,l z ~ r! , c n,l z ~ r! 5exp~ i w l z ! f n,l z ~ r ! , ~9!

S

D

H r 52

\ 2 2 e 2 B 2 2 e\B 1 1 ˆ ¹ 1 r 1 2 l 1V ~ r ! , 2m r 8mc2 2c m e m h i z ~10!

V~ r !5

E E dz e

dz h U eh~ r ,z e ,z h ! z 21 ~ z e ! j 21 ~ z h ! ,

~11!

@where m 5(1/m e 11/m h i ) 21 is the reduced exciton mass#, which can be easily integrated numerically. The generalization of this procedure to a multilevel situation ~e.g., for several QW levels or for coupled double or triple QWs! will be presented elsewhere.18 This approach ~the numerical solution of the in-plane magnetoexciton Schro¨dinger equation! produces the same results ~with absolute accuracy better than 0.1 meV! for the 1s,...4s exciton binding energies as the previously reported approaches in zero10 and in quantizing11 magnetic fields. It is also applicable in the intermediate magnetic fields. Thus, within the framework of the single approach it is possible to calculate the evolution of the exciton states in the whole range of magnetic fields.

FIG. 1. Landau fan diagrams of the NSQW magnetoexciton, calculated for L cap520 and 4 nm ~solid and dashed lines, respectively!. Dashed-dotted lines in the inset are a redshifted fan diagram for L cap55 nm so that the 1s transition energy coincides with that for L cap520 nm.

B. Magnetoexciton transition and binding energies

The calculated magnetoexciton transition energies of the lowest s and p states, \ v ns and \ v np , take the form of well known fan diagrams and are illustrated in Fig. 1. The numerical calculations were carried out for L QW55 nm In0.18Ga0.82As/GaAs NSQWs, using the following parameters: V e 52113 meV and V h 5275 meV, m ez5m e i 50.067, m hz50.35, m h i 50.2, and e 512.5. Left panel in Fig. 1 displays the fans, calculated for two cap layer thicknesses, L cap520 and 4 nm, at zero electric field E50. Note a small energy splitting between the 2s, 2 p 6 states at B 50, which is a consequence of the non-Coulombic form of the effective 2D exciton interaction, Eq. ~11!. It is seen that the decrease of the cap layer thickness causes a strong blueshift of all transition energies, \ v n . The blueshift increases with the level number, n, and decreases with increasing magnetic field. These changes can be seen better in the inset of Fig. 1, where we have shifted the calculated Landau fan for L cap54 nm QW so that its 1s transition energy at B50 coincides with that for L cap520 nm QW. The blueshift of magnetoexciton levels with decreasing L cap is due to two different reasons. The first one is the influence of the high vacuum potential barrier in a close vicinity of the QW. This barrier causes an additional ~to the QW! localization of electrons and holes. Hence, it enhances the free electron and free hole energies with the decrease of L cap . 19 This effect has a tunneling origin; we will call it a ‘‘tunneling blueshift.’’ The second reason for the blueshift is the repulsion between charges and their self-images, accounted for by the term V self , Eq. ~3!. We will call this component of the shift a ‘‘dielectric blueshift.’’ For an electrically neutral exciton, the dielectric blueshift is partly compensated by the attraction



Gippius et al.

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FIG. 3. Dependences of 1s exciton binding energies, E 1s 5\ v 002\ v 1s , on L cap , calculated with and without accounting for the interaction with images ~solid and dashed lines, respectively!, at B50 and 14 T ~thin and thick lines, respectively!.

FIG. 2. Dependences of the 1s transition energies, \ v 1s and \ v 1s,0 , on the cap layer thickness, L cap , calculated with and without accounting for the interaction with images ~thick solid and dashed lines, respectively!, at B 50, 7 and 14 T. Corresponding dependences of energy splittings between zero free electron and hole Landau levels ~LLs!, \ v 00 ~of band gap in the case of B50!, are shown by thin solid and dashed lines. The inset displays the magnetic field dependence of the dielectric blueshift, \ v 1s 2\ v 1s,0 , calculated for L cap53 nm.

between the hole ~electron! and the electron ~hole! image @the second term in Eq. ~5!#. This attraction is just the exciton dielectric enhancement. It enhances the exciton binding energies and thus reduces magnetoexciton blueshift with the decrease of L cap . Figure 2 shows the calculated dependences of \ v 1s ~thick solid curves!, in comparison with the dependences of the energy gap between zero Landau levels of the noninteracting electron and hole, \ v 00 ~thin solid curves!, for B 50, 7, and 14 T at E50. To distinguish between the tunneling and dielectric parts of the blueshift of the 1s state, we display in Fig. 2 as well the dependences \ v 1s and \ v 00 ~thick and thin dashed lines, respectively!, calculated without the terms, responsible for the interaction with the image charges, i.e., Eq. ~3! and the second term in Eq. ~5!. Physically, the corresponding Hamiltonian describes a system where vacuum is replaced by a dielectric with infinite interface potential barriers, but with the same dielectric constant as in the semiconductor. The difference between magnetoexciton transition energies \ v 1s , calculated with and without accounting for image charges, is much smaller than that for \ v 00 . Moreover, as the inset at the bottom panel of Fig. 2 shows, this difference decreases with B at a fixed L cap . Thus, the 1s transition energy blueshift has mainly the tunneling origin, and its small dielectric counterpart is further decreased with the increase of the magnetic field.

Figure 3 compares the dielectric enhancement of binding energy, E 1s 5 u \ v 002\ v 1s u , in the range of L cap52–20 nm at B50 and 14 T ~thin and thick solid lines, respectively!. In addition to a well known increase of E 1s in a magnetic field, Fig. 3 shows as well that E 1s increases strongly with approaching of the QW to vacuum.20 The latter dependence is completely due to interaction with images. The calculated E 1s without allowing for image charges ~shown in Fig. 3 by dashed curves! does not depend markedly on L cap . Thus the calculations show that the tunneling effect influences very weakly the exciton binding energy, but it causes a large 1s transition energy blueshift. In contrast, the dielectric effect is responsible for the exciton enhancement, but changes weakly 1s transition energy. Such a behavior originates from the exciton electroneutrality. The smaller the in-plane size of the magnetoexciton, the better is the cancelling ~in the exciton transition energy! of the dielectric blueshift of \ v 00 by the redshift due to the enhanced binding energy. The squeezing of magnetoexcitons in a magnetic field reduces the blueshift of exciton levels as shown in the inset in Fig. 2. As expected, the change of the transition energy due to the dielectric effect is larger for higher, more extended, magnetoexciton states. These effects are clearly seen in Fig. 1, right panel. Thus, to measure the dielectric confinement effect, the evolution of inter-level splittings, D n 8 n 5E n 8 s 2E ns rather than the interband transition energies has to be investigated. One more reason why the investigation of inter-level magnetoexciton splittings is preferable is connected to an influence of the surface charge induced electric field, which may exist in NSQWs ~see, e.g., Ref. 21!. The latter effect increases when the QW approaches vacuum. We have calculated the influence of the electric field on the magnetoexciton transition energies in NSQWs with strong vertical confinement, and have found that the perpendicular electric field E5(0,0,E) causes nearly the same Stark shifts of all lowerlying magnetoexciton levels. This is illustrated in Fig. 4, where the change of the transition energies E ns 2E ns u E50 , n51,2,3,4 ~top panel! and of the magnetoexciton level split-


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Gippius et al.


some magnitude of the electric field. Under positive electric fields ~when the SV interface has a positive net charge!, the exciton dipole moment grows up. III. EXPERIMENT A. Experimental setup

FIG. 4. Calculated Stark shift of the magnetoexciton transition energies, \ v ns 2\ v ns u E50 , n51,2,3,4 ~top!, and the change of the inter-level splittings, D n1 2D n1 u E50 , n52,3,4 ~bottom!, at B57 T and L cap55 nm.

tings D n1 2D n1 u E50 , n52,3,4 ~bottom panel! is shown as a function of the electric field E up to 650 kV/cm ~which corresponds to the SV interface charge density of '63 31011 cm22!. Figure 4 shows that the Stark shift of transition energies is nearly one order of magnitude larger than the change of the inter-level splittings. This is not surprising because the electric field directly influences the QW confined levels, whereas the effective in-plane e-h interaction in the QW exciton is changed only through the modification of the QW wave functions z 1 (z e ), z 1 (z h ). For a QW in the bulk, the Stark shift is quadratic in E, because the exciton wave function has a definite parity under inversion z→2z. In an NSQW this symmetry is broken, and the Stark effect becomes linear in a weak electric field as shown in Fig. 4, because of nonzero dipole moment of the exciton in a NSQW, directed from the NSQW into the bulk. The effect depends on L cap , and in principle can be used for experimental investigation of the magnitude and direction of the near-surface electric field. Physically, this dipole moment appears because of repulsion of the electron and hole from their self-images, and as a consequence of electron and hole mass difference: heavier holes localize inside a NSQW under this repulsion at a larger distance from the SV interface. ~Note that without accounting for images, a tunneling effect causes nonzero exciton dipole moment of the opposite direction, from QW towards the SV interface, because of vacuum barrier pushing out of lighter electrons.! Under a negative electric field ~see Fig. 4!, which corresponds to a negative net charge on the SV interface, the exciton dipole moment decreases. It comes through zero at

For the measurements we have chosen strained Inx Ga12x As/GaAs heterostructures with x;0.18 and a QW thickness L QW55 nm. At this thickness, there is only one bound state in the conduction band. A stress induced splitting of the valence band in the InGaAs layer exceeds 40 meV.12 Therefore any spectral features in this energy range are related to the excited states of the heavy-hole exciton (2s, etc!. The additional advantages of such QW structures are ~i! a strong ~one order of magnitude! decrease of e at the SV interface, and ~ii! a large exciton radius (a B ;80 Å). An as grown sample with L cap520 nm was used as a reference. After measuring photoluminescence ~PL! and photoluminescence excitation ~PLE! spectra, the cap layer was thinned by etching. Two problems have to be solved in the etching process, namely, one has to avoid surface defects and to avoid cap layer thickness fluctuations ~after the etching!. For example, fluctuations of only about 2 ML at L cap53 nm result in such a strong emission line broadening that PLE spectroscopy becomes impossible. We have used dry etching by an Ar1-ion beam to remove the cap layer gradually. To avoid defect formation, low ~500 eV! ion energy, low sample temperature ~liquid nitrogen!, and a small angle between the ion beam and the sample surface ~20°! have been used.22 Both PL and PLE spectra were recorded with the use of a Ti-sapphire laser and a double grating monochromator RAMANOR U1000 at 4.2 K. The sample was located in a helium cryostat with superconducting solenoid. The emission of the NSQW was excited and collected via the same quartz fiber with a diameter of 0.6 mm located just near ~0.2 mm! the sample surface. A cooled photomultiplier in the photon counting mode has been used for the detection of the PL. B. PL and PLE spectra of the NSQW

Typical PL and PLE exciton spectra are displayed ~by dashed and solid curves, respectively! in Fig. 5 for NSQWs with L cap520, 5, and 3 nm and at several magnetic fields B